Theorem eqelssd 3261

Description: Equality deduction from subclass relationship and membership. (Contributed by AV, 21-Aug-2022.)

Hypotheses

Ref	Expression
eqelssd.1	|- ( ph -> A C_ B )
eqelssd.2	|- ( ( ph /\ x e. B ) -> x e. A )

Assertion

Ref	Expression
eqelssd	|- ( ph -> A = B )

Distinct variable groups:   x, A   x, B   ph, x

Proof of Theorem eqelssd

Step	Hyp	Ref	Expression
1		eqelssd.1	. 2 |- ( ph -> A C_ B )
2		eqelssd.2	. . . 4 |- ( ( ph /\ x e. B ) -> x e. A )
3	2	ex 115	. . 3 |- ( ph -> ( x e. B -> x e. A ) )
4	3	ssrdv 3248	. 2 |- ( ph -> B C_ A )
5	1, 4	eqssd 3259	1 |- ( ph -> A = B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   C_ wss 3214

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-in 3220  df-ss 3227

This theorem is referenced by:  fiuni  7278  ennnfonelemrn  13254  ennnfonelemdm  13255  unirnblps  15413  unirnbl  15414  dvidlemap  15682  dvidrelem  15683  dvidsslem  15684  dviaddf  15696  dvimulf  15697  dvcj  15700  dvrecap  15704